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Weird if it contains exactly one of the distinct elements

Source: VAIMO 2007, P1

January 3, 2009
combinatorics unsolvedcombinatorics

Problem Statement

Let n>1,n∈Z n > 1, n \in \mathbb{Z} and B \equal{}\{1,2,\ldots, 2^n\}. A subset A A of B B is called weird if it contains exactly one of the distinct elements x,y∈B x,y \in B such that the sum of x x and y y is a power of two. How many weird subsets does B B have?
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